17.6.2. Mapping to Ordered Pairs

You are given arbitrary distinct points in 2-dimensional planes, each having integer coordinates. Our goal is to show that irrespective of how these 5 points are chosen, there always exists a pair of points such that their midpoint has integer coordinates.

In this part, we are dealing with 5 distinct points in a 2D space. Each point has integer coordinates which can be expressed as (x, y). The aim is to find two points from these five such that the midpoint of the line connecting them also has integer coordinates. The midpoint formula is:

Midpoint = ( (x1 + x2) / 2 , (y1 + y2) / 2 ).

For the midpoint to have integer coordinates, both (x1 + x2) and (y1 + y2) must be even, or both must be odd.

Imagine you are at a party with 5 friends who each have different colored shirts (the distinct points). You want to pair them based on the color combinations (even/odd). If you group them by shirt color, you can ensure that at least one group will share similarities allowing a unique pairing—just like we need to find points with matching coordinate properties.

We can categorize the points based on whether their x or y coordinates are odd or even. Thus, we have four combinations for each point: (even, even), (even, odd), (odd, even), and (odd, odd). This means we map our 5 points into 4 categories.

Here, the Pigeonhole Principle becomes crucial. We have 5 distinct points but only 4 categories based on their parity (even or odd). According to the principle, if you have more items (points) than containers (categories), at least one container must contain more than one item. Therefore, at least two of the 5 points must fall into the same category.

Think of it like having 5 different fruits but only 4 baskets to put them in. Since you have more fruits than baskets, at least one basket will end up holding 2 or more fruits. In our case, this is similar, where points are the fruits and parity categories are the baskets.

Assuming the two points that map to the same pair are (x1, y1) and (x2, y2). Since they belong to the same category, it follows that both x1 and x2 are either odd or even, and similarly, y1 and y2.

Since both points (x1, y1) and (x2, y2) are categorized similarly, their x-coordinates will either be both odd or both even. The same applies to the y-coordinates. Thus, the sums x1 + x2 and y1 + y2 will be even—making the resulting midpoint integer coordinate values.

Mathematically, if both x-coordinates are odd: (odd + odd) = even.

If both are even: (even + even) = even.

Imagine you are placing two pencils either both on a desk or both in a drawer. If you have two even lengths, they combine for an even total length when placed next to each other. Likewise, two pencils with odd lengths will still total an even length, making it neatly organized (integer coordinates).

Thus, it is established that among any 5 distinct points chosen in a 2-dimensional integer coordinate system, there will always be at least one pair of points whose midpoint has integer coordinates.

This conclusion directly follows from applying the Pigeonhole Principle and the facts about parity. The approach ensures that no matter how we select the 5 points, we can be confident in finding at least one pair that, when averaged, produces an integer result.

It's like having multiple guests at a dinner trying to select dessert. No matter which flavors are chosen (the points), you will always be able to find at least one pairing that perfectly satisfies everyone (the integer midpoint) because they all belong to the array of possible dessert choices.

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